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This paper concerns the multi-component coupled Kardar-Parisi-Zhang (KPZ)equation and its two types of approximations. One approximation is obtained asa simple replacement of the noise term by a smeared noise with a properrenormalization, while the other one introduced in [6] is suitable for studyingthe invariant measures. By applying the paracontrolled calculus introduced byGubinelli et al. [8, 9], we show that two approximations have the common limitunder the properly adjusted choice of renormalization factors for each of theseapproximations. In particular, if the coupling constants of the nonlinear termof the coupled KPZ equation satisfy the so-called "trilinear" condition, therenormalization factors can be taken the same in two approximations and thedifference of the limits of two approximations are explicitly computed.Moreover, under the trilinear condition, the Wiener measure twisted by thediffusion matrix becomes stationary for the limit and we show that the solutionof the limit equation exists globally in time when the initial value is sampledfrom the stationary measure. This is shown for the associated tilt process.Combined with the strong Feller property shown by Hairer and Mattingly [12],this result can be extended for all initial values.

A coupled KPZ equation, its two types of approximations and existence of global solutions